TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Third isomorphism theorem for rings: $(R / I) / (J / I) ≅ R / J$

Statement

Third isomorphism theorem for rings: $(R / I) / (J / I) ≅ R / J$ .

Facts & Assumptions

Given: Two-sided ideals $I \subseteq J ⊴ R$ .

[L1]

$J / I$ is an ideal of $R / I$ (If $I \subseteq J$ are ideals of $R$ , then $J / I$ is an ideal of $R / I$ ).

[L2]

A ring modulo a kernel is isomorphic to the image (First isomorphism theorem for rings: $R / \ker f ≅ im f$ ).

[L3]

Quotient projections are surjective ring homomorphisms (The canonical projection $R \to R / I$ is a surjective ring homomorphism with kernel $I$ ).

Proof

technique · direct

1.1

Define $ϕ : R / I \to R / J$ by $ϕ (r + I) = r + J$ ; it is a well-defined surjective ring homomorphism because $I \subseteq J$ .

L1L2L3givenconstruct

2.1

The equality $ϕ (r + I) = 0 + J$ holds exactly when $r \in J$ , so $\ker ϕ = J / I$ .

step 1.1L1L2L3givenalgebra

3.1

The kernel and image computation gives $(R / I) / (J / I) ≅ R / J$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Judson, Abstract Algebra: Theory and Applications, Ring Homomorphisms and Ideals (standard reference, not scraped)

Third isomorphism theorem for rings: (R/I)/(J/I)≅R/J(R/I)/(J/I)≅R/J